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c. The reflexive closure of a quasi order is a partial order.

d. Every finite poset has a minimal element and a maximal element

10. List the ordered pairs in the relation R from A = {0,1, 2, 3} to B = {0, 1, 2, 3, 4} where (a, b) R if and only if

a. a>b.

b. a + b = 3.

c. a divides b.

d. a - b = 0.

e. gcd( a , b ) = 1.

f. lcm( a , b ) = 6.

11. Recursively define the relation { ( a , b ) | a=2b }, where a and b are natural numbers.

12. List unary relation on {1, 2, 3}.

13. Prove that there are 2 n2 binary relations on a set of cardinality n .

14. For each of the following relations on the set {1, 2, 3, 4}, decide whether it is reflexive, symmetric, antisymmetric and/or transitive.

a. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

b. {(1, 3), (1, 4), (2, 3), (3, 4)}

c. {(1, 1), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 3), (3, 4)}

15. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where ( x , y ) ∈ R if and only if

a. x is divisible by y .

b. x y.

c. y = x + 2 or y = x - 2.

d. x = y ² + 1 .

16. Let A be the set of people in your town. Let R 1 be the unary relation representing the people in your town who were registered in the last election  and R 2 be the unary relation representing the people in your town who voted in the last election.  Describe the 1-tuples in each of the following relations.

a. R 1 ∪ R 2.

b. R 1 ∩ R 2.

17. Draw the directed graph that represents the relation {( a , b ), ( a , c ), ( b , c ), ( c , b ), ( c , c ), ( c , d ), ( d , a ), ( d , b )}.

18. Let R be the parent-child relation on the set of people that is, R = { ( a, b ) | a is a parent of b }.  Let S be the sibling relation on the set of people that is, R = { ( a , b ) | a and b are siblings (brothers or sisters) }.  What are S o R and R o S ?

19. Let R be a reflexive relation on a set A .  Show that R n is reflexive for all positive integers n .

20. Let R be the relation on the set { 1, 2, 3, 4} containing the ordered pairs (1, 1), (1, 2), (2, 2), (2, 4), (3, 4), and (4, 1).  Find

a. the reflexive closure of R

b. symmetric closure of R   and

c. transitive closure of R .

21. Let R be the relation { ( a, b ) | a is a (integer) multiple of b } on the set of integers.  What is the symmetric closure of R ?

22. Suppose that a binary relation R on a set A is reflexive.  Show that   R*   is reflexive, where   R* = i = 1 n R i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } {R} rSup { size 8{i} } } {} .

23. Which of the following relations on {1, 2, 3, 4} are equivalence relations?  Determine the properties of an equivalence relation that the others lack.

a. {(1, 1), (2, 2), (3, 3), (4, 4)}

b. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

c. {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}

24. Suppose that A is a nonempty set, and f is a function that has A as its domain.  Let R be the relation on A consisting of all ordered pairs ( x , y ) where f(x) = f(y) .

a. Show that R is an equivalence relation on A .

b. What are the equivalence classes of R ?

25. Show that propositional equivalence is an equivalence relation on the set of all compound propositions.

26. Give a description of each of the congruence classes modulo 6.

27. Which of the following collections of subsets are partitions of {1, 2, 3, 4, 5, 6}?

a. {1, 2, 3}, {3, 4}, {4, 5, 6}

b. {1, 2, 6}, {3, 5}, {4}

c. {2, 4, 6}, {1, 5}

d. {1, 4, 5}, {2, 3, 6}

28. Consider the equivalence relation on the set of integers R = { ( x, y ) | x - y is an integer}.

a. What is the equivalence class of 1 for this equivalence relation?

b. What is the equivalence class of 0.3 for this equivalence relation?

29. Which of the following are posets?

a. ( Z,  =  )

b. ( Z, ≠)

c. ( A collection of sets, ⊆).

30. Draw the Hasse diagram for the divisibility relation on the following sets

a. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

b. {1, 2, 5, 8, 16, 32}

31. Answer the following questions concerning the poset ({{1}, {2}, {3}, {4}, {1, 3}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {2, 3, 4}},⊆).

a. Find the maximal elements.

b. Find the minimal elements.

c. Is there a greatest element?

d Is there a least element?

e. Find all upper bounds of {{2}, {4}}.

f. Find the least upper bound of {{2}, {4}}, if it exists.

g. Find all lower bounds of {{1, 2, 4}, {2, 3, 4}}

h. Find the greatest lower bound of {{1, 2, 4}, {2, 3, 4}}, if it exists.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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